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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X18,12,5,11 X10,18,11,17 X2,14,3,13

Gauss Code:

{{1, -9, 5, -3}, {4, -1, 2, -5, 6, -8, 7, -4, 9, -2, 3, -6, 8, -7}}

Jones Polynomial:

q-3/2 - 3q-1/2 + 3q1/2 - 6q3/2 + 6q5/2 - 7q7/2 + 6q9/2 - 4q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

- q-4 + q-2 + 1 + 3q2 + 4q4 + 2q6 + 4q8 - q10 - 2q14 - 2q16 - q20 + q22

HOMFLY-PT Polynomial:

a-5z-1 - a-5z - 3a-5z3 - a-5z5 - 3a-3z-1 + a-3z + 7a-3z3 + 5a-3z5 + a-3z7 + 2a-1z-1 - 3a-1z3 - a-1z5

Kauffman Polynomial:

- a-9z3 + 2a-8z2 - 3a-8z4 + 3a-7z3 - 4a-7z5 - a-6 + 2a-6z2 + 3a-6z4 - 4a-6z6 + a-5z-1 + 2a-5z - 6a-5z3 + 8a-5z5 - 4a-5z7 - 3a-4 + 3a-4z4 + 2a-4z6 - 2a-4z8 + 3a-3z-1 + 3a-3z - 22a-3z3 + 24a-3z5 - 7a-3z7 - 3a-2 - a-2z2 + 5a-2z6 - 2a-2z8 + 2a-1z-1 + a-1z - 12a-1z3 + 12a-1z5 - 3a-1z7 - z2 + 3z4 - z6

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 16                   1 j = 14                 2   j = 12               2 1   j = 10             4 2     j = 8           3 2       j = 6         3 4         j = 4       3 3           j = 2     2 5             j = 0   1 1               j = -2   2                 j = -4 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[9, Alternating, 2]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 2]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[12, 6, 13, 5], > X[8, 4, 9, 3], X[16, 10, 17, 9], X[18, 12, 5, 11], X[10, 18, 11, 17], > X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 5, -3}, {4, -1, 2, -5, 6, -8, 7, -4, 9, -2, 3, -6, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(3/2) 3 3/2 5/2 7/2 9/2 11/2 q - ------- + 3 Sqrt[q] - 6 q + 6 q - 7 q + 6 q - 4 q + Sqrt[q] 13/2 15/2 > 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-4 -2 2 4 6 8 10 14 16 20 22 1 - q + q + 3 q + 4 q + 2 q + 4 q - q - 2 q - 2 q - q + q
In[8]:=	HOMFLYPT[Link[9, Alternating, 2]][a, z]
Out[8]=	3 3 3 5 5 5 7 1 3 2 z z 3 z 7 z 3 z z 5 z z z ---- - ---- + --- - -- + -- - ---- + ---- - ---- - -- + ---- - -- + -- 5 3 a z 5 3 5 3 a 5 3 a 3 a z a z a a a a a a a
In[9]:=	Kauffman[Link[9, Alternating, 2]][a, z]
Out[9]=	2 2 2 -6 3 3 1 3 2 2 z 3 z z 2 2 z 2 z z -a - -- - -- + ---- + ---- + --- + --- + --- + - - z + ---- + ---- - -- - 4 2 5 3 a z 5 3 a 8 6 2 a a a z a z a a a a a 3 3 3 3 3 4 4 4 5 z 3 z 6 z 22 z 12 z 4 3 z 3 z 3 z 4 z > -- + ---- - ---- - ----- - ----- + 3 z - ---- + ---- + ---- - ---- + 9 7 5 3 a 8 6 4 7 a a a a a a a a 5 5 5 6 6 6 7 7 7 8 z 24 z 12 z 6 4 z 2 z 5 z 4 z 7 z 3 z > ---- + ----- + ----- - z - ---- + ---- + ---- - ---- - ---- - ---- - 5 3 a 6 4 2 5 3 a a a a a a a a 8 8 2 z 2 z > ---- - ---- 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 -2 2 1 2 q 4 6 6 2 5 q + 3 q + ----- + t + ----- + - + ---- + 3 q t + 3 q t + 4 q t + 4 3 2 2 t t q t q t 8 2 8 3 10 3 10 4 12 4 12 5 14 5 > 3 q t + 2 q t + 4 q t + 2 q t + 2 q t + q t + 2 q t + 16 6 > q t

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